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User blog:Holomanga/Trigonometric and hyperbolic functions; complex numbers
Recommended Textbook: Mathematical Methods for Physics and Engineering; Riley, Hobson and Bence, Chapters 1 and 3 Okay so the first bullet point is Trigonometric and hyperbolic functions; complex numbers. This is all one bullet point, for some reason, though I guess it's somewhat understandable. Trigonometric Functions The two big trigonometric functions are sine and cosine, written as \sin \theta and \cos \theta . They are defined with reference to the unit circle, which is a circle of radius 1 centered on the origin. As you can see from the diagram, cosine is the length of the side adjacent to the angle and sine is the length of the side opposite the angle. The ratio of these two functions is called the tangent, written as \tan \theta = \frac{\sin \theta}{\cos \theta} . The reciprocals of these two functions are called secant, cosecant and cotangent, written as \cot \theta = \frac{1}{\sin \theta} , \csc \theta = \frac{1}{\cos \theta} , and \cot \theta = \frac{1}{\tan \theta} . Useful relations between these trigonometric functions are called identities. The useful ones to know are the angle addition formulae, which allow you to find the sine and cosine of the sum of two angles, as well as the related double-angle formula Also important is \sin^2 \theta + \cos^2 \theta = 1 , which follows from Pythagoras' theorem and the definition of the sine and cosine. By dividing through by \sin^2 \theta and \cos^2 \theta , we can get similar results for the secant, cosecant, tangent, and cotangent. The angle addition formulae are: * \sin \left(\alpha \pm \beta \right) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta * \cos \left(\alpha \pm \beta \right) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta To prove these, stare at the accompanying diagram. It's purely geometric, which is nice. I don't actually know how to prove these geometrically, and just rely on the complex exponential, which I'm going to tell you about soon. Lil' bit of foreshadowing and excitement for you. If we set \alpha = \beta , then we get the double-angle formulae, * \sin 2 \theta = 2 \sin \theta \cos \theta * \cos 2 \theta = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta + 1 = 1 - 2 \sin^2 \theta By adding together the expressions for \sin \left(\alpha + \beta \right) and \sin \left(\alpha - \beta \right) , we get \sin \left(\alpha + \beta \right) + \sin \left(\alpha - \beta \right) = 2 \sin \alpha \cos \beta , which relates the sum of two sinusoids to the product of two sinusoids. In a similar way, we can obtain * \sin \left(\alpha + \beta \right) - \sin \left(\alpha - \beta \right) = 2 \cos \alpha \sin \beta * \cos \left(\alpha + \beta \right) + \cos \left(\alpha - \beta \right) = 2 \cos \alpha \cos \beta * \cos \left(\alpha + \beta \right) - \cos \left(\alpha - \beta \right) = -2 \sin \alpha \sin \beta The inverse of these functions are called arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangen, written as \arcsin , \arccos , \arctan , \text{arccsc } , \text{arcsec } and \text{arccot } . Arcsine, for example, is defined such that, if x = \sin y , then \arcsin y = x , and the other inverse trigonometric functions are defined analogously. Because the inverse trigonometric functions aren't single-valued (e.g. sin \left( x \right) = sin \left( x + 2k\pi \right) ), a domain must be selected for them. The domain you choose depends on what you're trying to find the inverse of; typically, it will be something like between - \pi and \pi such that angles are positive. Complex Numbers Complex numbers are numbers of the form z = x + iy , where x is said the be the real part, written as \text{Re }z , and y is said to be the imaginary part, written as \operatorname{Im}z . The constant i is a special number called the imaginary unit with the property that i^2 = -1 . Other than that, complex numbers behave totally normally, and there's no need to worry about them a bit. Adding together two complex numbers is easy: z_1 + z_2 = x_1 + iy_1 + x_2 + iy_2 = x_1 + x_2 + i\left(y_1 + y_2\right) . In other words, the real and imaginary parts are just added together separately. Multiplying two complex numbers is also easy: z_1 z_2 = \left( x_1 + iy_1 \right)\left( x_2 + iy_2 \right) = x_1 x_2 - y_1 y_2 + i \left( x_1 y_2 + x_2 y_1 \right) . It's the same as multiplying together two normal sums of numbers, but keeping in mind that the square of the imaginary unit is -1. Something that's useful is to plot complex numbers in two-dimensional space, with the x-axis being the real part and the y-axis being the imaginary part. This gives you something called an Argand diagram, and it looks like in the accompanying picture. In two dimensions, another way of representing a position is in polar coordinates, where you have an angle from the origin and a distance from the origin. Given the definition of sine and cosine, we can see that we can write a complex number in the form z = r \left( \cos \phi + i \sin \phi \right) . r is called the modulus of the complex number, and \phi is called the argument of the number. We can write the \phi component of that expression all as a single function for brevity. Let's define a new function, \exp i \phi = cos \phi + i \sin \phi . Let's also pose that, for real values, this function acts the same as e^x . This, right now, is an arbitrary definition that we're allowed to make - we'll justify why this is the natural choice when we come to calculus. We will then also be able to prove that this acts the same as the exponential (in that it follows all the same identities), so could reasonably be written as e^{i\phi} . We can then write z = r \exp i \phi . What's interesting is that we can now write sine and cosine in terms of this newly defined function, the exponential function. We find that \sin \theta = \frac{\exp \left( i\theta \right) - \exp \left( -i \theta \right)}{2i} , and that \cos \theta = \frac{\exp \left( i\theta \right) + \exp \left( -i \theta \right)}{2} . These forms, along with the standard rules for manipulating exponentials, can be used to derive the trigonometric identities. Back to complex numbers. The modulus of a complex number is also written as | z | . When written in the form z = r e^{i\theta} , we have |z| = r , and when written in the form z = x + iy , we have |z| = \sqrt{x^2 + y^2} . The argument of a complex number is often written as \arg z , and we have \arg z = \theta . The argument can also be obtained from the cartesian representation, as \arg z = \operatorname{atan2}(y,x) . \operatorname{atan2} is defined in terms of the arctangent function, and is used because the arctangent is unable to distinguish between opposite directions - \arctan \frac{y}{x} = \arctan \frac{-y}{-x} , even though the arguments of the complex numbers are clearly different, and is as follows: : \operatorname{atan2}(y,x) = \begin{cases} \arctan(\frac y x) &\text{if } x > 0, \\ \arctan(\frac y x) + \pi &\text{if } x < 0 \text{ and } y \ge 0, \\ \arctan(\frac y x) - \pi &\text{if } x < 0 \text{ and } y < 0, \\ +\frac{\pi}{2} &\text{if } x = 0 \text{ and } y > 0, \\ -\frac{\pi}{2} &\text{if } x = 0 \text{ and } y < 0 \end{cases} Since this is the standard arctangent, possibly plus or minus \pi depending on the quadrant, it is usually possible to find \arctan(\frac y x) and then pick an appropriate value by inspection; plotting the complex number on an Argand diagram helps with this. A complex number z has an associated number called the complex conjugate, which is its reflection in the real axis. It is written as z^* and is equal to re^{-i\theta} , x - iy , or z - 2\operatorname{Im}z , depending on which form is most useful. Multiplying a complex number by its complex conjugate gives the square of its modulus: z z^* = \left( x + iy\right) \left( x - iy \right) = x^2 + y^2 = |z|^2 . Hyperbolic Functions Given that the trigonometric functions can be written in terms of the imaginary exponential, it might be interesting to see if analogous functions exist in terms of the real exponential. The answer is that they do - these functions are called the hyperbolic functions. We have the hyperbolic sine, defined analogously to the sine as \sinh x = \frac{e^x - e^{-x}}{2} , and the hyperbolic cosine, defined analogously to the cosine as \cosh x = \frac{e^x + e^{-x}}{2} . We can then define the hyperbolic tangent \tanh x = \frac{\sinh x}{\cosh x} , and the hyperbolic secant, cosecant and cotangent as \text{sech } x = \frac{1}{\cosh x} , \text{csch } x = \frac{1}{\sinh x} , and \coth x = \frac{1}{\tanh x} . All these hyperbolic functions are plotted. These functions hold some identities similar to those for trigonometric functions. Most importantly is \cosh^2 x - \sinh^2 x = 1 , which is analogous to the Pythagorean trigonometric identity and can be proven by expanding the hyperbolic functions out as exponentials. This identity also justifies why they are called the hyperbolic functions. On a circle, x^2 + y^2 = 1 , and since \sin^2 t + \cos^2 t = 1 , a circle can be parametrised by x = \cos t and y = \sin t . On a hyperbola, however, x^2 - y^2 = 1 and therefore the appropriate parametrisation to use is x = \cosh t and y = \sinh t . Category:Blog posts